Chern characters for supersymmetric field theories

We construct a map from $d|1$-dimensional Euclidean field theories to complexified K-theory when $d=1$ and complex analytic elliptic cohomology when $d=2$. This provides further evidence for the Stolz--Teichner program, while also identifying candidate geometric models for Chern characters within their framework. The construction arises as a higher-dimensional and parameterized generalization of Fei Han's realization of the Chern character in K-theory as dimensional reduction for $1|1$-dimensional Euclidean field theories. In the elliptic case, the main new feature is a subtle interplay between the geometry of the super moduli space of $2|1$-dimensional tori and the derived geometry of complex analytic elliptic cohomology. As a corollary, we obtain an entirely geometric proof that partition functions of $\mathcal{N}=(0,1)$ supersymmetric quantum field theories are weak modular forms, following a suggestion of Stolz and Teichner.


Geometry & Topology msp 1 Introduction and statement of results
Given a smooth manifold M , Stolz and Teichner [26] have constructed categories of dj1-dimensional super Euclidean field theories over M for d D 1; 2, We elaborate on this connection between Theorem 1.1 and the conjectures (2).The maps (3) and ( 4) come from sending a field theory to its partition function.This assignment defines a type of character map for field theories.Similarly, the cohomology theories in (2) have Chern characters valued in certain cohomology theories defined over C. Putting these ingredients together, we obtain the diagrams One expects the cocycle maps in (2) will make these diagrams commute.This offers new perspective on the conjectures (2), as we briefly summarize.Extending a partition function to a full field theory requires both additional data and property: a choice of preimage under the map restr in (3) and (4) need not exist nor be unique.Similarly, refining a cohomology class over C to a class in the target of ( 2) is both data and property: a class is in the image of the Chern character if it satisfies an integrality condition, and lifts of integral classes need not be unique owing to the presence of torsion.Up to an equivalence relation called concordance (see below), the conjectures (2) assert that the data and property determining such refinements -either as field theories or cohomology classes -precisely match each other.
The concordance relation features in the full conjecture of Stolz and Teichner, which asserts that the cocycle maps (2) induce bijections between concordance classes of field theories and cohomology classes.Recall that for a sheaf F W Mfld op !Set on the site of manifolds, sections s 0 ; s 1 2 F.M / are concordant if there exists s 2 F.M R/ such that s 0 D i 0 s and s 1 D i 1 s, where i 0 ; i 1 W M ,! M R are the inclusions at 0 and 1.
Concordance defines an equivalence relation on the set F.M /, whose equivalence classes are concordance classes.There is an analogous definition of concordance for (higher) stacks, where the stack condition is used to show that the concordance relation is transitive.Assuming that M 7 !dj1-EFT.M / is a d-stack, Proposition 1.2 implies that concordance classes of dj1-dimensional Euclidean field theories map to H.M I COEˇ; ˇ 1 / and H.M I MF/ for d D 1 and 2, respectively.We expect this to implement the Chern character for K-theory and TMF through the maps on concordance classes induced by the diagrams (5).This brings us to a technical point: although it is expected that the assignment M 7 !d j1-EFT.M / is a d-stack, when d D 2 this statement is contingent on a fully extended enhancement of the existing definitions.This fully extended aspect is an essential ingredient in Stolz and Teichner's conjecture that concordance classes of 2j1dimensional field theories yield TMF; see [26,Conjecture 1.17].In this paper, the source of (4) uses the 1-categorical definition from [26].Fully extended 2j1-dimensional super Euclidean field theories should map to this 1-categorical version (via a forgetful functor), and from this one would obtain a Chern character on concordance classes via postcomposition with (4).

Cocycles from partition functions
In physics, the best-known topological invariants associated with the field theories (1) are the Witten index in dimension 1j1 (see eg Witten [27]), and the elliptic genus in dimension 2j1 (see eg Witten [28] or Alvarez, Killingback, Mangano and Windey [1]).These are examples of partition functions.For example, when d D 2 the partition function of the N D .0;1/ supersymmetric sigma model with target a string manifold is a modular form called the Witten genus; see Witten [29].This genus led Segal [23] to suggest that certain 2-dimensional quantum field theories could provide a geometric model for elliptic cohomology.
Stolz and Teichner refined these early ideas, leading to the conjectured cocycle maps (2).In their framework (as in Segal's [24]), partition functions are defined as the value of a field theory on closed, connected bordisms [26,Definition 4.13].The definition of a super Euclidean field theory implies that this restriction determines a function invariant under the action by super Euclidean isometries (6) dj1-EFT.M / !C 1 .fclosedbordisms over M g/ isometries : Fei Han [18] shows that (6) applied to a class of 1j1-dimensional closed bordisms over M , (7) encodes the Chern character in K-theory.To summarize, restriction along (7) evaluates a 1j1-dimensional Euclidean field theory on length 1 super circles whose map to M is invariant under the action of loop rotation.This restriction is also a version of dimensional reduction.When the input 1j1-dimensional Euclidean field theory is constructed via Dumitrescu's [14] super parallel transport for a vector bundle with connection, the resulting element in C 1 .Map.R 0j1 ; M // ' .M / is a differential form representative of the Chern character of that vector bundle.
The cocycle map ( 4) is a more elaborate version of restriction along (7).The goal is to find an appropriate class of closed 2j1-dimensional bordisms so that the restriction (6) constructs a map from 2j1-dimensional Euclidean field theories to complex-analytic elliptic cohomology.There are two main problems to be solved in this 2-dimensional generalization.First, one cannot specialize to a particular super torus, as in the specialization to the length 1 super circle in (7).Indeed, elliptic cohomology over C is parametrized by the moduli of all complex-analytic elliptic curves.This problem is easy enough to solve, though its resolution introduces some technicalities: one restricts to a moduli stack of super tori.
The second obstacle is more serious.(iii) Construct the cocycle maps (3) and (4) using the output of step (ii).

Outline of the proof
The main work is in step (ii), culminating in Propositions 1.4 and 1.5 below.
For step (i), we start by defining where M dj1 is the moduli space of super Euclidean structures on R dj1 =Z d , and Map.R dj1 =Z d ; M / is the generalized supermanifold of maps from R dj1 =Z d to M .Hence, an S-point of L 1j1 .M / determines a family of super Euclidean circles with a map to M , and an S-point of L 2j1 .M / determines a family of super Euclidean tori with a map to M .There is a canonical functor L d j1 .M / !dj1-Bord.M /, regarding these supermanifolds as bordisms from the empty set to the empty set.Next we consider the subobject of (8) gotten by taking maps invariant under the R d -action on R d j1 =Z d by precomposition.Equivalently, this is the S 1 D R=Z-fixed subspace when d D 1 and the T 2 D R 2 =Z 2 -fixed subspace when d D 2. This yields finite-dimensional subobjects that, roughly speaking, are the subspaces of maps that are constant up to nilpotents.Restricting a field theory along the composition L dj1 0 .M / L dj1 .M / !d j1-Bord.M / extracts a function, providing the first arrow in (3) and ( 4), (10) restrW See Lemmas 2.12 and 3.15.

Remark 1.3
The restriction (10) is dimensional reduction in the sense of [10,Glossary], though it differs from dimensional reduction in the sense of [26,Section 1.3].
Step (ii) is a technical computation.The d D 1 case is characterized as follows.
Proposition 1.4 The elements of C 1 .L 1j1 0 .M // Euc 1j1 are in bijection with pairs .Z; Z `/, where Here Z is closed of total degree zero, Z `is of total degree 1, and they satisfy where d is the de Rham differential on M, and @ `is the vector field on R >0 associated to the standard coordinate `2 C 1 .R >0 /.
For the d D 2 case, let H C denote the upper half-plane with standard complex coordinates ; x 2 C 1 .H/, and let v 2 C 1 .R >0 / be the standard coordinate.
Proposition 1.5 The elements of C 1 .L 2j1 0 .M // Euc 2j1 are in bijection with triples .Z; Z x ; Z v /, where Here Z is closed of total degree zero, Z x and Z v are of total degree 1, they satisfy an SL 2 .Z/-invariance property stated in Lemma 3.23, and where d is the de Rham differential on M, and @ x and @ v are vector fields on H and R >0 .
In Propositions 1.4 and 1.5, the closed differential form Z arises by restriction to a subspace where Lat is the space of based, oriented lattices in C. Indeed, (11)  ; Z v /; using the description of functions from Propositions 1.4 and 1.5 and the maps on coefficients described in the previous two paragraphs.The definition of the maps (18) and (19) together with the de Rham theorem then implies that every cohomology class admits a refinement to a function on L The following remarks relate our results to other work.Remark 1.6 The above analysis of the moduli space of super Euclidean tori is related to previous investigations of moduli spaces of super Riemann surfaces in the string theory literature; see eg Donagi and Witten [13] and Witten [30].However, the vast majority of prior constructions in string theory and in the Stolz-Teichner program only study the reduced moduli spaces.In particular, the cocycle models for (equivariant) elliptic cohomology in Berwick-Evans [7; 6], Barthel, Berwick-Evans and Stapleton [3] and Berwick-Evans and Tripathy [8] arise as functions on the reduced moduli space.In this prior work, the correct mathematical object comes only after imposing holomorphy by hand.However, as Theorem 1.1 shows, this property emerges naturally from the geometry of 2j1-dimensional super tori.
Remark 1.7 When M D pt, Proposition 1.5 shows that partition functions of N D .0;1/ supersymmetric quantum field theories are weak modular forms: odd .pt/D f0g, so Z v D Z x D 0 are no additional data.In contrast to the arguments in the physics literature that analyze a particular action functional (eg [9,), the proof here emerges entirely from the geometry of the moduli space of super Euclidean tori.This recovers Stolz and Teichner's claim from [26, page 10] that "holomorphicity is a consequence of the more intricate structure of the moduli stack of supertori".
Remark 1.8 The data Z x in Proposition 1.5 is closely related to anomaly cancellation in physics and choices of string structures in geometry.An illustrative example is the elliptic Euler class: an oriented vector bundle V !M determines a class OEEu.V / 2 H.M I MF/ if the Pontryagin class OEp 1 .V / 2 H 4 .M I R/ vanishes.In Section 3.7 we show that the set of differential forms H 2 3 .MI R/ with p 1 .V / D dH parametrizes cocycle refinements of OEEu.V / to a function on L 2j1 0 .M /.Geometrically, H is part of the data of a string structure on V .In physics, H is part of the data for anomaly cancellation in a theory of V -valued free fermions.Under the conjectured cocycle maps (2), V -valued free fermions are expected to furnish representatives of elliptic Euler classes in TMF.M /; see Stolz and Teichner [25,Section 4.4].Perturbative quantization of fermions rigorously constructs elliptic Euler cocycles over C (see Berwick-Evans [6, Section 6]), and Theorem 1.1 shows that lifting a cohomology class to a 2j1-dimensional Euclidean field theory must depend on a choice of string structure, at least rationally.
Remark 1.9 If the input field theory in (4) is super conformal, then dZ v D 0, whereas if the input theory is holomorphic then dZ x D 0. For a general field theory (not necessarily conformal or holomorphic) the differential form @ x Z ` @ `Zx is closed.These closed forms have the potential to encode secondary cohomological invariants of field theories.Although we do not know explicit field theories for which this cohomology class is nonzero, the structure appears to be related to mock modular phenomena and the TMF-valued torsion invariants studied in Gaiotto, Johnson-Freyd and Witten [17] and Gaiotto and Johnson-Freyd [16].
Remark 1.10 In light of Fei Han's work [18] on the Bismut-Chern character, it is tempting to think of the restriction 2j1-EFT.M / !C 1 .L 2j1 .M // Euc 2j1 (without taking T 2 -invariant maps) as a candidate construction of the elliptic Bismut-Chern character.Indeed, functions on C 1 .L 2j1 .M // Euc 2j1 can be identified with cocycles analogous to (14), where Z is a differential form on the double loop space and the Daniel Berwick-Evans de Rham differential d is replaced with the T 2 -equivariant differential investigated in Berwick-Evans [5].

Conventions for supermanifolds
This paper works in the category of supermanifolds with structure sheaves defined over C; this is called the category of cs-supermanifolds in Deligne and Morgan [12].
The majority of what we require is covered in the concise introduction [26, Section 4.1], but we establish a little notation presently.First, all functions and differential forms are C-valued.The supermanifolds R njm are characterized by their super algebra of functions The representable presheaf associated with R njm assigns to a supermanifold S the set R njm .S / WD ft 1 ; t 2 ; : : : ; t n 2 C 1 .S/ ev ; Â 1 ; Â 2 ; : : where .ti / red denotes the restriction of a function to the reduced manifold S red ,! S, and .ti / red is the conjugate of the complex-valued function .ti / red on the smooth manifold S red .We use this functor of points description throughout the paper, typically with Roman letters denoting even functions and Greek letters denoting odd functions.
We follow Stolz and Teichner's terminology, wherein a presheaf on supermanifolds is called a generalized supermanifold.An example of a generalized supermanifold is Map.X; Y / for supermanifolds X and Y , which assigns to a supermanifold S the set of maps S X !Y .For a manifold M regarded as a supermanifold, the generalized supermanifold Map.R 0j1 ; M / is isomorphic to the representable presheaf associated to the odd tangent bundle …TM , as we recall briefly.We use the notation .x;/ 2 …TM.S/ for an S -point, where x W S !M is a map and 2 .SI x TM / odd is an odd section.This gives an S-point .xC Â / 2 Map.R 0j1 ; M / by identifying x with an algebra map x W C 1 .M / !C 1 .S/ and W C 1 .M / !C 1 .S/ with an odd derivation relative to x.These fit together to define an algebra map with the isomorphism coming from Taylor expansion in a choice of odd coordinate .M / recover differential forms on M as a Z=2-graded C-algebra.The action of automorphisms of R 0j1 on this algebra encode the de Rham differential and the grading operator on forms; see eg [19,Section 3].
Acknowledgements It is a pleasure to thank Bertram Arnold, Theo Johnson-Freyd, Stephan Stolz, Peter Teichner, and Arnav Tripathy for fruitful conversations that shaped this work, as well as a referee for finding a mistake in a previous draft and making several suggestions that improved the clarity and precision of the paper.

A map from 1j1-Euclidean field theories to complexified K-theory
The main goal of this section is to prove Proposition 1.4.From the discussion in Section 1.2, this proves the d D 1 case of Theorem 1.1.We also prove Proposition 1.2 when d D 1, and connect this result with Chern characters of super connections.The super Lie algebra of E 1j1 is generated by a single odd element, namely the leftinvariant vector field D D @ Â i Â @ t .The right-invariant generator is

The moduli space of super Euclidean circles
The super commutators are (24) 23) and ( 24) come from Wick rotation; see eg [12,  does not.Remark 2.5 There is a more general notion of a family of super circles where (26) incorporates the action by Z=2 < Euc 1j1 .This moduli space has two connected components corresponding to choices of spin structure on the underlying ordinary circle, with the component from Definition 2.3 corresponding to the odd (or nonbounding) spin structure.This turns out to be the relevant component to recover complexified K-theory.
We recall [26, Definitions 2.26, 2.33 and 4.4]: for a supermanifold M with an action by a super Lie group G, an .M; G/-structure on a family of supermanifolds with the ' i and satisfying a cocycle condition.An isometry between supermanifolds with .M; G/-structure is defined as a map T !T 0 over S that is locally given by the G-action on M, relative to the open covers fU i g of T and fU 0 i g of T 0 .Supermanifolds with .M; G/-structure and isometries form a category fibered over supermanifolds.Proof We endow a family of super circles with a 1j1-dimensional Euclidean structure as follows.Take the open cover S R 1j1 !S 1j1 `; supplied by the quotient map, and take transition data from the Z-action on S R 1j1 .By definition this Z-action is through super Euclidean isometries, and so the quotient inherits a super Euclidean structure.
We observe that every family of super circles pulls back from the universal family .R are the moduli space of super Euclidean circles and the universal family of super Euclidean circles, respectively.The following shows that M 1j1 D R respectively.Hence (28) determines a map between the respective Z-quotients, defining a map (27).This is easily seen to be an isomorphism of supermanifolds.Since (27) is not locally determined by the action of Euc 1j1 on R 1j1 , it is not a super Euclidean isometry.
The following result gives an S-point formula for the action of Euc 1j1 on S 1j1 and M 1j1 D R 1j1 >0 coming from isometries between super Euclidean circles.Remark 2.11 Precomposition actions (such as the action of Euc 1j1 on Map.S 1j1 ; M / above) are most naturally right actions.Turning this into a left action involves inversion on the group: the formula for 0 in (32) involves and the inverse of f .This inversion introduces signs in the formulas for the left Euc 1j1 -action on L 1j1 0 .M / below.Our choice to work with left actions is consistent with Freed's conventions for classical supersymmetric field theories [15, pages 44-45]; see also [11, page 357].
There is an evident S 1 -action on L 1j1 .M / coming from the precomposition action of S 1 D E=Z < E 1j1 =Z on Map.S 1j1 ; M /.Since the quotient is given by S 1j1 =S 1 ' R 0j1 , the S 1 -fixed points are (33) We identify an S-point of L 1j1 0 .M / with a map S 1j1 `; !M that factors as where the map p is induced by the projection R 1j1 !R 0j1 .The action (32) preserves this factorization condition; we give an explicit formula in Lemma 2.13 below.Hence, the inclusion (33) is Euc 1j1 -equivariant.
Lemma 2.12 There is a functor L

Computing the action of
which is the claimed formula for .x 0; 0 /.
Just as R-actions on ordinary manifolds are determined by flows of vector fields, E 1j1 -actions on supermanifolds are determined by the flow of an odd vector field.This comes from differentiating a left E 1j1 -action at zero and considering the action by the element Q of the super Lie algebra, using the notation from (24).Odd vector fields on supermanifolds are precisely odd derivations on their functions.We note the isomorphism Proof We recall that right-invariant vector fields generate left actions, so that the infinitesimal action of E 1j1 on L 1j1 0 .M / is determined by the action of Q.Furthermore, minus the de Rham operator generates the left E 0j1 -action .x;/ 7 !.xÁ ; / on …TM , and minus the degree derivation generates the left R -action .x;/ 7 !.x;u 1 /; see eg [19,Section 3.4].Applying the derivation Q D @ Á C i Á@ s to (36) and evaluating at .s;Á/ D 0 recovers (39).

The proof of Proposition 1.4
The Euc 1j1 -equivariant inclusion is along S -families of super circles with D 0. So by Lemmas 2.13 and 2.14 we have ' ev cl .M I C 1 .R >0 // using (36) to see that Z=2 acts through the parity involution (so invariant functions are even forms) and (39) to see that the E 1j1 -action is generated by minus the de Rham d (so invariant functions are closed forms).This verifies the equality (17)  Next, observe that where the final isomorphism comes from Taylor expansion of functions on R 1j1 >0 in the odd coordinate function .For convenience we choose the parametrization of functions We again have that Z=2 < Euc 1j1 acts by the parity involution, so since is odd and `is even we find where in the first equality we use that d.`dBy Lemma 2.14, y Q generates the E 1j1 -action and, since E 1j1 is connected, y Q-invariant functions are equivalent to E 1j1 -invariant functions.Finally, we identify even differential forms with elements of .M I C 1 .R >0 /OEˇ; ˇ 1 / of total degree zero and odd differential forms with elements of .M I C 1 .R >0 /OEˇ; ˇ 1 / of total degree 1 (essentially replacing `in (40) by ˇ).This completes the proof of Proposition 1.4.

Concordance classes of functions
For Proposition 1.2 we require a refinement of the cocycle map.

The Chern character of a super connection
A super connection A on a Z=2-graded vector bundle V !M is an odd C-linear map satisfying the Leibniz rule [22] AW .M I V / !.M I V /; A.f s/ D df s C .1/ jf j f As; for f 2 .M / and s 2 .M I V /.One can express a super connection as a finite sum A D P j A j , where A j W .M I V / !Cj .M I V / raises differential form degree by j .Note that A 1 is an ordinary connection on V , and A j is a differential form valued in End.V / odd if j is even and End.V / ev if j is odd.Super parallel transport provides a functor, denoted by sPar, from the groupoid of Z=2-graded vector bundles with super connection on M to the groupoid of 1j1-dimensional Euclidean field theories over M : .V; A/ 7 !sPar.V; A/ 7 !sTr.e `A2 /: Part of this construction is given in [14], reviewed in [ The R >0 -family of super connections (44) appears frequently in index theory, eg [22] and [4,  Remark 2.17 If A D r is an ordinary connection, the family (44) is independent of ànd Z `D 0. This recovers Fei Han's identification [18] of the Chern form Tr.exp.r 2 // with dimensional reduction of the 1j1-dimensional Euclidean field theory sPar.V; r/.

A map from 2j1-Euclidean field theories to complexified elliptic cohomology
The main goal of this section is to prove Proposition 1.5.From the discussion in Section 1.2, this proves Theorem 1.1 when d D 2. We also prove Proposition 1.2 when d D 2 and comment on connections with a de Rham model for complex-analytic elliptic cohomology, complexified TMF, and elliptic Euler classes.
Daniel Berwick-Evans

The moduli space of super Euclidean tori
We will use the two equivalent descriptions of S-points of R 2j1 : for .z;x z; Â/; .z 0; x z 0 ; Â 0 / 2 R 2j1 .S /.Define the super Euclidean group as E 2j1 ÌSpin.2/,where the semidirect product is defined by the action (using the notation (48)) .u; x u/ .z;x z; Â / D .u 2 z; x u 2 x z; x uÂ / for .u;x u/ 2 Spin.2/.S /: The Lie algebra of E 2j1 has one even generator and one odd generator.In terms of left-invariant vector fields, these are @ z and D D @ Â Â @ x z , whereas in terms of right-invariant vector fields they are @ z and Q D @ Â C Â @ x z .The super commutators are (50) OE@ z ; D D 0; OED; D D @ x z and OE@ z ; Q D 0; OEQ; Q D @ x z : for .n;m/ 2 Z 2 .S / and .z;x z; Â / 2 R 2j1 .S /.Equivalently, this is the restriction of the left E 2j1 -action on S R 2j1 to the S-family of subgroups S Z 2 S E 2j1 with generators over S specified by .`1;x `1; 1 / and .`2;x `2; 2 /.Define the standard super torus as T 2j1 D R 2j1 =Z 2 for the quotient by the action for the standard inclusion Z 2 R 2 E 2j1 , ie for the square lattice.

Remark 3.5
The S-family of subgroups S Z 2 ,! S E 2j1 determined by ƒ (as in Remark 3.3) is normal if and only if 1 D 2 D 0. Hence, although the standard super torus T 2j1 inherits a group structure from E 2j1 , generic super tori T 2j1 ƒ do not.Remark 3.6 There is a more general notion of a family of super tori where the action (52) also incorporates pairs of elements in Spin.2/.This moduli space has connected components corresponding to choices of spin structure on an ordinary torus, with the component from Definition 3.4 corresponding to the odd (or periodic-periodic) spin structure.This turns out to be the relevant component of the moduli space to recover complex-analytic elliptic cohomology.Every family of super tori pulls back from the universal family .sLatR 2j1 /=Z 2 !sLat along a map S !sLat.Hence, we regard as the moduli space of super Euclidean tori and the universal family of super Euclidean tori, respectively.The following identifies sLat with the moduli space of super Euclidean structures on the standard super torus.
Lemma 3.9 There exists an isomorphism of supermanifolds over sLat, from the constant sLat-family with fiber the standard super torus to the universal family of super Euclidean tori.This isomorphism does not preserve the super Euclidean structure on T 2j1 .
We will require an explicit description of functions on sLat, ie the morphisms of presheaves sLat !C 1 .Regarding Lat as a representable presheaf on supermanifolds, there is an evident monomorphism Lat ,! sLat from the canonical inclusion C C ' R 2 R 2 ,! R 2j1 R 2j1 .In the following, let 1 ; 2 2 C 1 .sLat/denote the restriction of the odd coordinate functions Daniel Berwick-Evans Lemma 3.12 There is an isomorphism of algebras where, as usual, we write the associated pair of maps S !R 2j1 as .`1;x `1; 1 / and .`2;x `2; 2 /.We therefore have 4 even and 2 odd functions on sLat that, as maps of sheaves sLat !C 1 , assign to an S -point the functions `1; x `1; `2; x `2 2 C 1 .S/ ev or 1 ; 2 2 C 1 .S/ odd .It is easy to see that arbitrary smooth functions in the variables `1; x `1; `2; x `2 continue to define maps of sheaves and hence smooth functions on sLat.Furthermore, since these are the restriction of functions on R 2 R 2 R 2j1 R 2j1 , we can identify them with functions on Lat.This specifies the even subalgebra C 1 .Lat/ C 1 .sLat/.On the other hand, the odd functions 1 and 2 are subject to a relation coming from condition (i) in Definition 3.2, namely that 1 2 D 2 1 2 C 1 .S/ odd for all S. Since these are odd functions, this is equivalent to the condition that 1 2 D 0. Hence the functions on sLat are as claimed.

Remark 3.13
The relation 1 2 D 0 implies that C 1 .sLat/ is not the algebra of functions on any supermanifold, and hence the generalized supermanifold sLat fails to be representable.where the horizontal arrows are the inverses of the isomorphisms of supermanifolds pulled back from Lemma 3.9, and f is the super Euclidean isometry associated to an S-point of Euc 2j1 in Lemma 3.11.These isomorphisms together with the arrow uniquely determine 0 in (58).Hence, for .ƒ;/ 2 sLat.S / Map.T 2j1 ; M /.S / and an S-point of Euc 2j1 , we define the Euc 2j1 -action on L 2j1 .M / as outputting .ƒ 0; 0 / in (58).We caution that this is a left Euc 2j1 -action on sLat Map.T 2j1 ; M /, and refer to Remark 2.11 for a discussion of left actions on mapping spaces.

Super Euclidean double loop spaces
There is a T 2 -action on L 2j1 .M / coming from the T 2 -action on Map.T 2j1 ; M / by the precomposition action of T 2 on T 2j1 .The T 2 -fixed points comprise the subspace (59) We identify an S-point of this subspace as a map where the map p is induced by the projection R 2j1 !R 0j1 .The action (58) preserves this factorization condition; we give explicit formulae in Lemma 3.17 below.Hence, the inclusion ( 59) is Euc 2j1 -equivariant.

Computing the action of super Euclidean isometries
Definition 3.16 Using the notation from Lemma 3.12, define the function The restriction of vol along Lat ,! sLat is the function that reads off the volume of an ordinary torus C=`1Z ˚`2 Z using the flat metric.In particular, this function is real-valued, positive and invertible.By Lemma 3.12, the function vol on sLat is also invertible.
Lemma 3.17 The left E 2j1 Ì Spin.2/-action on sLat Map.R 0j1 ; M / is given by (63) .w;x w; Á; u; x u/ .`1;x `1; 1 ; `2; x `2; 2 ; x;  where the arrow labeled by f .w;x w;Á;u;x u/ denotes the associated map between super Euclidean tori from Lemma 3.11.For the first statement in the present lemma we take D id 2 SL 2 .Z/.S/.We see that ƒ 0 is given by (55).To compute .x 0; 0 /, we find a formula for the dashed arrow in (64) that makes the triangle commute.To start, part of the data of the inverse to the isomorphism (54) is (65) We verify that z p ƒ is Z 2 -invariant for the action (52), z p `; ..n; m/ .z; where we used (62).Hence z p ƒ determines a map p ƒ W T 2j1 ƒ !S R 0j1 , which is the map in (64).From this we see that the dashed arrow in (37) is unique and determined by (66) As in Remark 2.11, the left action of E 2j1 Ì Spin.2/ on .xC Â / 2 Map.R 0j1 ; M /.S/ is given by which gives the claimed formula for .x 0; 0 /.Finally, a short computation shows that p ƒ D p ƒ 0 ı , where W T 2j1 ƒ !T 2j1 ƒ 0 is the isometry associated to 2 SL 2 .Z/.S/ from Lemma 3.11.Hence, the SL 2 .Z/-action on sLat Map.R 0j1 ; M / is indeed through the action on sLat.
From the Lie algebra description (50), a left E 2j1 -action determines an even and an odd vector field gotten by considering the infinitesimal action by the elements Q D @ Â CÂ @ x z and @ z of the Lie algebra of E 2j1 .We note the isomorphisms where in (67) we used that the projective tensor product of Fréchet spaces satisfies where d is the de Rham differential and deg is the degree endomorphism on forms.
Proof The proof follows the same reasoning as the proof of Lemma 2.14, using that right-invariant vector fields generate left actions and that the E 0j1 Ì C -action on Map.R 0j1 ; M / is generated by minus the de Rham operator and the degree derivation.

3.4
The proof of Proposition 1.5 where d is the de Rham differential on M .
Geometry & Topology, Volume 27 (2023) Proof The element 1 2 U.1/ ' Spin.2/ acts through the parity involution, which on C 1 .sLat/ is determined by i 7 !i .Using (69) and the fact that vol is an invertible function on Lat, we see that any Z=2-invariant function can be written in the form (70). Next we compute for !
Matching coefficients of 1 and 2 , the condition y Q! D 0 is therefore equivalent to (71).Finally, invariance under the operator y @ w from Lemma 3.18 follows from being y Q-closed, specifically from d! 0 D 0. Since E 2j1 is connected with Lie algebra generated by y Q and y @ w , we find that (71) completely specifies the subalgebra Next we compute the Spin.2/-invariantfunctions.Consider the surjective map (72) ' W Lat !H R >0 ; .`1;x `1; `2; x `2/ 7 !.`1=`2;x `1= x `2; vol/ 2 .H R >0 /.S/; and use the pullback on functions to get an injection We observe that the image of this map is precisely determined by the map (73) on coefficients, where !0 is in the image of an element of total degree zero and ! 1 ; ! 2 are in the image of elements of total degree 1.
The following allows us to recast the invariance condition as a failure of Z D ! 0 to have holomorphic dependence on the conformal modulus and be independent of volume.and the inclusion on the right regards a weakly holomorphic modular form as a weak modular form.We expect the image of 2j1-Euclidean field theories along (4) to satisfy this meromorphicity property at i 1, and hence have image in the subring TMF.M / ˝C.This follows from an "energy bounded below" condition discussed for M D pt in [26,Section 3].However, proving that the image of field theories satisfies this condition requires that one analyze the values of field theories on super tori and super annuli.

Concordance classes of functions
The cocycle map (4) can be factored through a complex that computes the derived global sections of the elliptic cohomology sheaf, namely the complex ..M I 0; .H/OEˇ; ˇ 1 / SL 2 .Z/ ; d C x @/ described above.Proof Let us verify that the map in Definition 3.27 is well-defined.By Proposition 1.5, the image is contained in the subspace of degree zero cocycles: .d C x @/.Z./ C d x Z x .// D d x @ x Z. / d x dZ x ./ D 0: The image is SL 2 .Z/-invariant by Lemma 3.23.The remainder of the proof is completely analogous to that of Lemma 2.16.When dim V D 24k, we may ask for a preimage of k OEEu.V / 2 H 0 .M I MF/ under the cocycle map (4), where is the modular discriminant.We start with the differential form refinement of Eu.V /, evident from its definition above, Eu.V / 2 .M I C 1 .H/OEˇ; ˇ 1 /; @ x Eu.V / D ˇ2Tr.F 2 / 4 i.
x / 2 Eu.V /; and whose failure to be holomorphic is as indicated.Since @ v Eu.V / D 0, we may choose Z D k Eu.V / and Z v D 0. The remaining data to promote k Eu.V / to a function on L 2j1 0 .M / is a choice of coboundary @ x .k Eu.V // D dZ x , which in turn is determined by H 2 3 .M / with dH D p 1 .V /, ie a rational string structure.

( 1 ) 2 . 1 . 1
d j1-EFT.M / WD Fun ˝.d j1-EBord.M /; V/: Its objects are symmetric monoidal functors from a bordism category dj1-EBord.M / to a category of vector spaces V.The morphisms of dj1-EBord.M / are dj1-dimensional super Euclidean bordisms with a map to a smooth manifold M .For details we refer to Stolz and Teichner [26, Section 4].In [26, Sections 1.5-1.6],they conjectured the existence of cocycle maps 1948 Daniel Berwick-Evans circles with maps to M when d D 1, and super tori with maps to M when d D 2, both viewed as a particular class of closed bordisms over M .A super Lie group Euc dj1 acts through super Euclidean isometries on super circles and super tori, inducing actions on L dj1 0 .M / for d D 1; Theorem The invariant functions C 1 .L d j1 0 .M // Euc d j1 determine cocycles in 2periodic cohomology with complex coefficients when d D 1, and cohomology with coefficients in the ring MF of weak modular forms when d D 2. Composing with restriction along L d j1 0 .M / dj1-EBord.M / determines maps from field theories to these cohomology theories over C: (3) 1j1-EFT.M / restr !C 1 .L 1j1 0 .M // Euc 1j1 cocycle H.M I COEˇ; ˇ 1 / with jˇj D 2, and (4) 2j1-EFT.M / restr !C 1 .L 2j1 0 .M // Euc 2j1 cocycle H.M I MF/: For M D pt, the map (4) specializes to part of an announced result of Stolz and Teichner [26, Theorem 1.15]; see Remark 3.24.Applied to general manifolds M , one can identify H.I COEˇ; ˇ 1 / with complexified K-theory, and H.I MF/ with a version of TMF over C; see Section 3.5.Hence, Theorem 1.1 proves a version of the conjectures (2) over C.

Proposition 1 . 2
The assignment M 7 !C 1 .L dj1 0 .M // Euc d j1 is a sheaf on the site of manifolds.Concordance classes of sections map surjectively to H.M I COEˇ; ˇ 1 / and H.M I MF/ when d D 1 and 2, respectively.

Theorem 1 .
1 boils down to somewhat technical computations in supermanifolds, so we briefly outline the approach and state key intermediate results in terms of ordinary (nonsuper) geometry.There are three main steps in the construction: (i) Construct the super moduli spaces L d j1 0 .M /. (ii) Compute the algebras of Euc dj1 -invariant functions C 1 .L d j1 0 .M // Euc d j1 in terms of differential form data on M .
page 95, Example 4.9.3].This differs from the convention for the 1j1dimensional Euclidean group in[20, Definition 33], but is more closely aligned with the Wick rotated 2j1-dimensional Euclidean geometry defined in [26, Section 4.2] and studied below.Let R 1j1 >0 denote the supermanifold gotten by restricting the structure sheaf of R 1j1 to the positive reals, R >0 R. Definition 2.3 Given an S -point .`;/ 2 R 1j1 >0 .S /, the family of 1j1-dimensional super Euclidean circles is defined as the quotient (25) S 1j1 `; WD .S R 1j1 /=Z for the left Z-action over S determined by the formula (26) n .t;Â / D .tC n`C i n Â; n C Â / for n 2 Z.S/; .t;Â / 2 R 1j1 .S /: Equivalently this is the restriction of the left E 1j1 -action on S R 1j1 to the S-family of subgroups Z S E 1j1 S with generator f1g S ' S .`;/ , !R 1j1 >0 S E 1j1 S: Define the standard super Euclidean circle, denoted by S 1j1 D S 1j1 1;0 D R 1j1 =Z, as the quotient by the action for the standard inclusion Z R E 1j1 .Remark 2.4 The S -family of subgroups S Z ,! S E 1j1 generated by .`;/ 2 R 1j1 >0 .S/ is normal if and only if D 0. Hence, the standard super circle S 1j1 inherits a group structure from E 1j1 , but a generic S -family of super Euclidean circles S 1j1 `; Definition 2.6 [26, Section 4.2] A super Euclidean structure on a 1j1-dimensional family T !S is an .M; G/-structure for the left action of G D Euc 1j1 on M D R 1j1 .Lemma 2.7 An S-family of super circles (25) has a canonical super Euclidean structure.

1j1 0 .
M / !1j1-EBord.M / that induces a restriction map (35) restrW 1j1-EFT.M / !C 1 .L 1j1 0 .M // Euc 1j1 : Geometry & Topology, Volume 27 (2023) Daniel Berwick-Evans Proof The 1j1-dimensional Euclidean bordism category over M is constructed by inputting the 1j1-dimensional Euclidean geometry from Definition 2.6 into the definition of a geometric bordism category [26, Definition 4.12].The result is a category 1j1-EBord.M / internal to stacks on the site of supermanifolds; in particular, 1j1-EBord.M / has a stack of morphisms consisting of proper families of 1j1dimensional Euclidean manifolds with a map to M , with additional decorations related to the source and target of a bordism.By Lemma 2.7, super Euclidean circles give examples of S-families of 1j1-dimensional Euclidean manifolds.An S-point of L 1j1 0.M / therefore defines a proper S-family of 1j1-Euclidean manifolds with a map to M via (34).We can identify this with an S-family of morphisms in 1j1-EBord.M / whose source and target are the empty supermanifold equipped with the unique map to M .This defines a functor L 1j1 0 .M / !1j1-EBord.M / and a restriction map 1j1-EFT.M / !C 1 .L 1j1 0 .M //.We refer to the discussion preceding [26, Definition 4.13] for an explanation why the restriction to closed bordisms extracts a function from a field theory.Finally we argue that this restriction has image in Euc 1j1 -invariant functions.By definition, an isometry between 1j1-dimensional Euclidean manifolds comes from the action of the super Euclidean group Euc 1j1 D E 1j1 Ì Z=2 on the open cover defining the super Euclidean manifold.By Lemma 2.9, the action (32) on L 1j1 0 .M / is therefore through super Euclidean isometries of super circles compatible with the maps to M .By definition, these isometries define isomorphisms between the bordisms (34) in 1j1-EBord.M /.Functions on a stack are functions on objects invariant under the action of isomorphisms.Hence, the restriction 1j1-EFT.M / !C 1 .L 1j1 0 .M // necessarily takes values in functions invariant under Euc 1j1 , yielding the claimed map (35).
OE ˝ .M /; Geometry & Topology, Volume 27 (2023) where (in an abuse of notation) we let `; 2 C 1 .R 1j1 >0 / denote the coordinate functions associated with the universal family of super circles S D R 1j1 >0 !R 1j1 >0 R 1j1 .In the above, we used that C 1 .Map.R 0j1 ; M // ' .M / and C 1 .S T / ' C 1 .S/ ˝C 1 .T / for supermanifolds S and T using the projective tensor product of Fréchet algebras; see for instance [20, Example 49].Let degW .M / !.M / denote the (even) degree derivation determined by deg.!/ D k! for ! 2 k .M /.Lemma 2.14 The left E 1j1 -action (36) on L 1j1 0 .M / is generated by the odd derivation (39) y Q WD 2i d d`˝i d id ˝d i `˝deg using the identification of functions (38), where d is the de Rham differential and deg is the degree derivation on differential forms.
when d D 1 and extracts the data Z from an element of C 1 .L 1j1 0 .M // Euc 1j1 .Geometry & Topology, Volume 27 (2023) Stolz and Teichner's .M; G/-structures are discussed before Definition 2.6.Definition 3.7 [26, Section 4.2] A super Euclidean structure on a 2j1-dimensional family T !S is an .M; G/-structure for the left action of G D E 2j1 Ì Spin.2/ on M D R 2j1 .Lemma 3.8 An S-family of super tori (51) has a canonical super Euclidean structure.Proof The proof is the same as for Lemma 2.7, using the open cover S R 2j1 !T 2j1 ƒ and transition data from the Z 2 -action (52).

Definition 3 . 14 0
Define the super Euclidean double loop space as the generalized supermanifold L 2j1 .M / WD sLat Map.T 2j1 ; M /: We identify an S -point of L 2j1 .M / with a map T 2j1 ƒ !M given by the composition (57) T 2j1 ƒ ' S T 2j1 !M; using the isomorphism from Lemma 3.9.We shall define a left action of Euc 2j1 on L 2j1 .M / determined by the diagram (58) Geometry & Topology, Volume 27 (2023)

Proof of Proposition 1 .2 for d D 2
By Proposition 1.5, M 7 !C 1 .L 2j1 0 .M // Euc 2j1 is a sheaf on the site of smooth manifolds.The map in Definition 3.27 is a morphism of sheaves2 cocycle W C 1 .L 2j1 0 .// Euc 2j1 !Z 0 . .I 0; .H/OEˇ; ˇ 1 /; d C x @/ SL 2 .Z/ :When evaluated on a manifold M , concordance classes of sections of the target are cohomology classes.This completes the proof.
Stolz andTeichner's field theories are neither chiral nor conformal, and hence restriction only gives a smooth function on the moduli stack of super Euclidean tori.On the other hand, a class in complex-analytic elliptic cohomology only depends on the holomorphic part of the conformal modulus of a torus.Resolving this apparent mismatch comes through a surprising feature of the super moduli space L 2j1 0 .M / possess a kind of derived holomorphy and conformality.
26, Section 1.3].A different approach (satisfying stronger naturality properties required to construct the functor sPar) is work in progress by Arnold [2].Evaluating the field theory sPar.V; A/ on closed bordisms determines the function sTr.e `A2 / 2 C 1 .R >0 Map.R 0j1 ; M //.The parametrization (40) extracts the function Z determined by `deg =2 Z D sTr.exp.`A 2 //: Hence we find that Z D sTr.exp.A 2 `// for (44) Euc 1j1 refining the Chern character of the Z=2-graded vector bundle V .
Definition 3.2 Define the generalized supermanifold of based (super) lattices in R 2j1 as the subfunctor sLat R 2j1 R 2j1 (viewing R 2j1 R 2j1 as a representable presheaf) whose S-points are .`1;x`1; 1 /; .`2;x`2; 2 / 2 R 2j1 .S / such that:(i) 2j1 ƒ WD .S R 2j1 /=Z 2 for the free left Z 2 -action over S determined by the formula (52) .n;m/ .z;x z The following gives an S-point formula for the action of Euc 2j1 on T 2j1 and M 2j1 D sLat coming from isometries between super Euclidean tori.
[26,a 3.21 An E 2j1 Ì Spin.2/-invariant function on L Z x ; Z v / where Z 2 .M I C 1 .H R >0 /OEˇ; ˇ 1 / has total degree zero and Z v ; Z x 2 .M I C 1 .H R >0 /OEˇ; ˇ 1 / have total degree 1 and satisfy Remark 3.24 As announced in[26, Theorem 1.15], a 2j1-Euclidean field theory over M D pt has a partition function valued in integral modular forms.Theorem 1.1 when d D 2 specializes to the holomorphy and modularity statements in this result when M D pt; generalizing the integrality statement would require one to consider the values of field theories on super annuli with maps to M .Remark 3.25The Lie groupoid Lat==Spin.2/SL 2 .Z/ gives a presentation of the moduli stack of Euclidean tori with periodic-periodic spin structure and choice of basepoint, where SL 2 .Z/ Spin.2/ acts via the restriction of the action from Lemma 3.17.The involution generated by 1 2 U.1/ ' Spin.2/ is the spin flip automorphism, which acts trivially on the underlying Euclidean torus and by the parity involution on the spinor bundle.Consider the subspace H R >0 Lat of based lattices whose second generator `2 2 R >0 C is positive and real.Since every based lattice can be rotated to one of this form (using the action of Spin.2/ on Lat) the full subgroupoid of Lat==Spin.2/SL 2 .Z/ with the objects H R >0 Lat is equivalent to Lat==Spin.2/SL 2 .Z/.Since f˙1g Spin.2/ acts trivially on the subspace H R >0 Lat, the manifold of morphisms in this full subgroupoid is H R >0 f˙1g SL 2 .Z/.Composition of morphisms gives the set f˙1g SL 2 .Z/ the structure of a group, which turns out to be the metaplectic double cover MP 2 .Z/ of SL 2 .Z/.There is a functor between Lie groupoids uW H R >0 ==MP 2 .Z/ !H==MP 2 .Z/, where the target is a standard presentation for the stack of complex-analytic elliptic curves endowed with a periodic-periodic spin structure.Geometrically, the functor u extracts the underlying complex-analytic elliptic curve with spin structure.Finally, observe there is a functor Lat==Spin.2/SL 2 .Z/ !M 2j1 ==Euc 2j1 , so a family of Euclidean tori with spin structure and choice of basepoint determines a family of super tori.Our arguments involving super tori do not encounter the metaplectic double cover because at the outset (in Lemma 3.19) we restrict to functions invariant under the spin flip automorphism.Hence only the quotient MP 2 .Z/=f˙1g ' SL 2 .Z/ features in our arguments.3.5 Weak modular forms and complexified TMF Definition 3.26 Weak modular forms of weight k are holomorphic functions f 2 O.H/ satisfying Let MF k denote the C-vector space of weak modular forms of weight k.Define the graded ring of weak modular forms MF as the graded vector space Cohomology with coefficients in weak modular forms is the object that naturally appears when studying derived global sections of the elliptic cohomology sheaf in the complex-analytic context.Indeed, complex-analytic elliptic cohomology assigns to a smooth manifold M a sheaf E``.M / of differential graded algebras on the orbifold H==SL 2 .Z/ with values (79) E``.M /.U / WD .O.U I .M /OEˇ; ˇ 1 /; d/ for U H: The SL 2 .Z/-equivariance data for this sheaf comes from pulling back functions along fractional linear transformations and sending ˇ7 !.cC d/ˇ.This connects with standard definitions of elliptic cohomology in homotopy theory (eg [21, Definition 1.2]) by identifying H==SL 2 .Z/ with the moduli stack of complex-analytic elliptic curves, and values (79) with the de Rham complex for 2-periodic cohomology with coefficients in O.U /.Using the Dolbeault resolution of holomorphic functions on H, the complex ..M I 0; .H/OEˇ; ˇ 1 / SL 2 .Z/ ; d C x @/ computes the derived global sections (ie the hypercohomology) of the elliptic cohomology sheaf E``.M /.Since H is Stein, the inclusion O.H/ ,! .0;.H/; x @/ is a quasi-isomorphism.Hence, derived global sections of the elliptic cohomology sheaf are cohomology with values in weak modular forms, H.M I O.H/OEˇ; ˇ 1 / SL 2 .Z/ ' H.M I MF/:We refer to [5, Section 3] for details.A weak modular form is a weakly holomorphic modular form if it is meromorphic as! i 1.For M compact, cohomology with values in weakly holomorphic modular forms is isomorphic to the complexification of topological modular forms, v Z D dZ v and @ x Z D dZ x for coordinates .; x / on H and v on R >0 .Geometry & Topology, Volume 27 (2023) Daniel Berwick-Evans f Â a C b c C d Ã D .cC d/ k f ./ for 2 H; (80) TMF.M / ˝C ' H.M I TMF.pt/ ˝C/ H.M I MF/; TMF.pt/ ˝C ' fweakly holomorphic modular formsg MF; Geometry & Topology, Volume 27 (2023) 3.7 The elliptic Euler class as a cocycleFor a real oriented vector bundle V !M , consider the characteristic class/ 2k Tr.F 2k / Ã in H dim V .M I C 1 .H/OEˇ; ˇ 1 / SL 2 .Z/ , where F D r ı r 2 2 .M I End.V //is the curvature for a choice of a metric-compatible connection r on V and Pf.ˇR/ is the Pfaffian.The functions E 2k 2 C 1 .H/ are the 2k th Eisenstein series, where we take E 2 to be the modular, nonholomorphic version of the second Eisenstein series, E 2 .; x / D lim C m/ 2 jn C mj 2 ; E 2 .; x / D E hol 2 ./ whose relationship with the holomorphic (but not modular) second Eisenstein series E hol 2 ./ is as indicated.For k > 1, the Eisenstein series E 2k 2 O.H/ are holomorphic.Thus, if OEp 1 .V / D OETr.F 2 /=.2.2 i / 2 / 2 H 4 .M I R/ vanishes, then OEEu.V / 2 H dim V .M I O.H/OEˇ; ˇ 1 / SL 2 .Z/ is a holomorphic class.